Alumina nanostructures and three families of aluminoxanes, linear, cyclic, and cagelike structures, have structures that resemble their isovalent electronic hydrocarbon analogues. Specific examples of each family are the counterparts of fullerene, allene, benzene, and cubane, respectively. The aluminoxanes and alumina nanostructures are related to each other; the latter can be regarded as a hydrogen- or alkyl-free form of aluminoxane. By exploiting this relationship, the relative stabilities of the three families of aluminoxanes, alumina nanostructures, and alumina crystal lattices have been estimated. According to ab initio calculations, aluminoxane cages, which take the form of a truncated octahedron and related polyhedra, are favored. The stability of the preferred cage, T-symmetric Al28O28H28, is practically equal to that of the alpha-alumina crystal lattice.